Macromolecules 1991,24, 2912-2919
2912
Colloid Stabilization with Grafted Polyelectrolytes P. Pincus Materials Department, University of California, Santa Barbara, California 93106 Received Auggust 20,1990; Revised Manuscript Received December 5,1990 ABSTRACT We consider the structure of a surface with a fairly dense array of polyelectrolytes end-graftad to it. A simple theory for the scaling properties of the corona thickness, counterion distribution, and disjoining pressure between opposing layers is presented, which makes contact with previous numerical studies. We show that by distributingthe charges on a macromoleculethe interlayer force is considerablyless sensitive to Debye screening by added electrolytes. This provides the physical basis for the importance of polyelectrolytes in colloid control in polar solvents. The flat layer studies are extended to the case of small (spherical) colloidal particles. Finally some speculations are made concerning the role of grafted polyelectrolytes on contributing to the bending energies of flexible membranes.
I. Introduction The stabilization of colloids by adsorbed' and endgrafted polymers2 in nonpolar solvents has been well investigated both experimentally and theoretically in recent years. In this situation, a generally accepted picture for the effects of macromolecules on modifying the forces between solid surfaces is used. In polar solvents, ionic macromolecules are often used for the control of colloidal properties. On the other hand, fundamental understanding of this situation has still not become established. There have been a few force balance experiments with adsorbed polyelectrolyte^.^ The Wageningen group has carried out some numerical studies of the mean-field equations4 in this case. More recently, two groupss*6have theoretically investigated polyelectrolytes that are end-grafted to solid Surfaces. These studies provide numerical solutions of the electrostatic mean-field equations (Boltzmann-Poisson equation) coupled to polymer elasticity for polyions grafted to flat surfaces. The aim of this work is to provide simple analytic scaling laws that are approximations to these results and to give extensions to the case of spherical colloidal particles. Finally we speculate on the contributions of grafted polyions to the curvature elasticity of flexible membranes. It would seem at f i t that, in polar solvents where charge separation is possible, it is sufficient to attach ionic groups to the surface of the particles that need to be kept from flocculating, sedimenting, or creaming. Indeed charged lattices' even form ordered colloidal crystals under the influence of the repulsive Coulombic interactions between the charged surfaces.e Although,in principle, this method is effective, in practice, it is not often used because of the extreme sensitivity to dissolved salts or other electrolytes. The range of the repulsive interaction is given by the Debye screening length K - ~ where ~2 = 4 m l ; n is the concentration of dissolved monovalent charged ions, and the Bjerrum length, 1, is given by 1 = e2/tT where e is the electronic charge, e is the dielectric constant of the solvent, and T is the temperature in energy While there is no formal difficulty in including multivalent species in the Boltzmann-Poisson theory, the screening lengths become sufficientlyshort that the point ion approximation is seriously in error.lOJ1 For dilute monovalent moieties, the point ion approximation seems to be, at least semiquantitatively, reasonable. (For water where the Bjerrum length is approximately 6 A, a 1 mM electrolyte corresponds to a Debye length of about 100 A. Thus even modest concentrations of salt provide strong screening of the repulsive interparticle Coulombic interaction.) As a 0024-9297/91/2224-2912$02.50/0
rough guideline,12 the range of the repulsive interaction should scale with the particle dimension in order to stabilize colloids against aggregation under the influence of van der Waals attractions. Since many applications of colloids involve situations where it is difficult to control ionic strengths at the millimolar level, it is necessary to find methods that are less sensitive to Debye screening. This is the principal motivation for using macromolecules. With charged macromolecules the stabilization is achieved with a combination of electrostatic and steric interactions. The polyions used may be moderate-sized ionic surfactants such as oleic acid or higher molecular weight polyelectrolytes. In this paper, we discuss the particular case of polyelectrolytes that are end-grafted to a solid surface. The end grafting may be achieved by covalently bonding the polymer to the surface with a chemically active end group. Prepolymerized molecules may thus be attached to an appropriately prepared surface, or the polymerization may even be initiated and carried out in situ. Alternately, the grafting topology may be obtained by adsorbing block copolymers with one block (possibly even nonpolar) that adsorbs on the surface, with a polyelectrolyte as the other block, e.g., polystyrene/ polystyrenesulfonate. Such copolymers may be simply viewed as macrosurfactants. Despite the importance of polyelectrolytes to colloid and emulsion control in aqueous solvents, why has relatively little work been done in this area? Part of the reason may be understood by contrast with our present level of knowledge concerning neutral polymers in nonpolar solvents at interfaces. In this case12-16our understanding of the surface properties relies on the foundation of the excellent level of current theory of semidilute polymer solution^.'^ On the other hand, in spite of some efforts,lgN bulk polyelectrolyte solutions have eluded any well-established theoretical picture that unifies the available empirical data. This lacuna has significantly retarded the development of the theory of charged polymers at interfaces. Our point of view, as well as that of the previous related studies of Miklavic and MarEelja6 and Misra, Varanasi, and Varanasi6 is that, for at least the case of end-grafted polyions, the existing familiarity with charged double-layer surfaces obviates a detailed understanding of the properties of polyelectrolyte solutions. Let us begin by reviewing the Boltzmann-Poisson theory for charged flat surfaces in the absence of polymers.21 Consider a flat solid surface of charge (negative) density 2-1 in contact with a polar solvent, e.g., water. We assume that the neutralizing monovalent counterions are arranged in the solvent 0 1991 American Chemical Society
Macromolecules, Vol. 24, No. 10, 1991
Colloid Stabilization with Grafted Polyelectrolytes 2913
according to the Boltzmann distribution. At this point, we specialize the case of no added electrolyte and neglect the contribution of water ionization to the Debye screening. The Boltzmann-Poisson (BP) equation is the mean-field description of the spatial counterion distribution. It is obtained by combinin Poisson’s equation of electrostatics, div B = 4uep where is the electric displacement, with the Boltzmann weighting for the counterion density, i.e., p = p ~ * * /where ~ p~ is a constant determined by global is the potential at the point i. charge neutrality and @(i) It is convenient to use dimensionless variables: 4 E e@/T for the potential and z = x/X for the length scale where x is the perpendicular distance from the surface into the solvent and the Guoy-Chapman length is A Z/2al. In terms of these variables, the BP equation is
%
d24/dz2= 4ulX2pQe” (1.1) This second-order nonlinear differential equation is solved by the counterion distribution p = [2al(x
+ X)2]-1
(1.2) Note that the counterions are effectivelylocalized to within a distance X from the charged surface. Indeed, it is useful to consider the counterions as an ideal gas confined to a sheath of thickness on the order of the Guoy-Chapman length. The Guoy-Chapman length is itself the Debye screening length associated with the mean counterion density in the sheath. Given the counterion distribution, properties such as the contribution of the double-layer interactions to the disjoining pressure between two identical parallel surfaces separated by a distance 2h may be easily derived.21 The physics of this situation may be simply understood by noting that the midplane between the surfaces is a symmetry plane on which the electric field must vanish. Therefore, the only contribution to the pressure across this plane arises from the mixing entropy of the counterions. If h >> A, this disjoining pressure, ll,is then given by II = Tp(h) = T/27rlh2. For h < A, the pressure becomes II E T/Zlh. In both limits, the pressure effectively competes with the attractive van der Waals force per unit area, which is given byz1 IIvw= -A/(4h3) where A is the Hamaker constant, which depends on the contrast in the dielectric function between the solids and the solvent and is quite generally on the order of T, i.e., an ambient thermal energy. Thus charged surfaces are stabilizing against aggregation and flocculation under the influence of dispersion attractive forces. However, as previously mentioned, this electrostatically induced repulsive interaction is easily screened by dissolved electrolytes. The use of ionic macromolecules greatly reduces this sensitivity, effectively coverting the exponential screening to a weaker power law behavior. As we shall see, this effect is associated with the entropic polymer elasticity. In order to understand the role of elasticity for macromolecules grafted to interfaces, we shall briefly review the principal points for the situation of end-grafted neutral polymers in contact with a good s~lvent.’~-’~ The planar geometry is sketched in Figure 1. We consider the simple case of monodisperse polymers of degree of polymerization N with a mean grafting density22 d-2 and with an excluded volume u per monomer. The polymers are extended in the form of a brush of thickness L, which is determined by balancing the excluded-volume repulsion tending to swell the chains against the entropic polymer elasticity. In mean-field theory, the osmotic pressure P corresponding to a local monomer concentration c is P = (1/2)uc2T. If a Gaussian polymer has an end-bend length
+
4
L
Figure 1. Sketch of the structure of a relatively dense grafted polymer brush. The brush height is L.
L, it stores an elastic energy (1/2)kL2,where the elastic constant k for a random-walk chain is k = (Na2)-lT(a is a monomer dimension). Then, assuming that all the grafted chains are stretched an identical length L, static equilibration is governed by
P = kL/d2 (1.3) If the monomers are uniformly distributed through the brush region, c = N(Ld2)-’. The equilibrium condition, (1.3), then determines the brush thickness to be L = Na(~/2ad~)’/~ (1.4) This result is consistent with Alexander’s14more sophisticated scaling discussion. The disjoining pressure between opposing identical brushes is exponentially small for h > L but becomes equal to the appropriate osmotic pressure P when the brushes are pushed against one another. Then
-
II = P = (1/2)uc2T uPT/(d4/h2) (1.5) This brush-induced pressure is of longer range23than the attractive van der Waals force, and, thus, if the grafting density is sufficiently large, it provides a barrier against adhesion in the dispersion-mediated minimum near contact. Subsequently, H i r numerically ~ ~ ~ solved the mean-field equations for this situation, but relaxing the assumption of a flat monomer distribution and the constraint that all free ends be at the same distance, L, from the surface. She found a parabolic profile but with the overall brush thickness obeying the Alexander scaling structure (1.4). This was rationalized analytically by Milner, Witten, and Cates (MWC)lSin terms of a WKB approximation to the Edwards analogy between polymers and the motion of a quantum mechanical particle.” Much of the physics of the Hirzian profile may be understood in terms of the following simple argument.26 The mean-field free energy per unit area, y, of a grafted layer may be expressed as (1.6) The first two terms in the integral, respectively, represent the excluded-volume energy and the polymer stretching energy, where * ( x ) is the concentration of free ends at x from the surface; P is a Lagrange multiplier to fix N monomers per chain. The problem is to determine \k(x).
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A correct approach is that given by MWC.lS However, a crude approximation may be obtained by guessing that W x ) = c(x)/N,i.e., the concentration of free ends scales with the monomer concentration. This is a reasonable approximati~nl~ in the outer extremities of the brush but quite poor near the surface. However, since the energetics is controlled by the highly extended chains, this ansatz gives quite good results. Now setting Gy/dc(x) = 0, we obtain
end concentration is given by c(x)/N, as is done in the Introduction for uncharged grafted chains; (iii) by a simple force balance assuming the counterions form a constrained ideal gas. The first two methods are carried out in the Appendix and serve to justify the force balance approach discussed in this section. As we shall see later, the results are also consistent with those of Miklavic and MarEelja.6 No Added Electrolyte. We first consider the situation in the absence of added electrolyte, i.e., the negatively charged polyions grafted to the surface with their monovalent neutralizing small counterions, e.g., Na+. As is clear from the discussion of the PB equation for a charged where p = [(3/2~~)(u/d2u)l2/3and the brush thickness is interface in the Introduction, the dominant effect of u ) ' / ~ . electrostatics is to preserve local charge neutrality at the consistent with Alexander scaling, L = N u ( ~ u / ~ ~The free energy associated with this profile is intermediate expense of the entropy of mixing of the counterions, which between the Alexander14and MWC16results. The overall is increased if they can explore larger volumes. Defining Hirzian parabolic form is clearly evident. 5 as the distance scale over which a test charge is neutralized,m we identify two possible regimes, namely L In the next section, we combine the PB equation with 1 5 and L I5. In fact, the latter one corresponds to most the approximate schemes for treating neutral grafted physical situations. Assuming that L 1 [, the brush polymers to obtain analytic expressions for the thickness thickness is determined by a balance between the swelling and forces between polyelectrolyte brushes. Section 111 effect of the counterion entropy and the chain elasticity. is devoted to an extension of these ideas to the case of In this regime the role of the electrostatics is to establish polyelectrolytes grafted to small spherical particles. Fithe neutralization length 5 but is not explicitly involved nally the concluding remarks include some speculations in determining the global brush thickness because neuon the contribution of grafted polyelectrolytes to the trality is quite local. Then, L may be found in a similar curvature elasticity of flexible membranes. manner to the derivation of (1.4), i.e., using P = kL/d2 where now P is the counterion osmotic pressure P = fcT, 11. Polyelectrolytes Grafted to Flat Surfaces where we have used the local neutrality condition to In this section, we consider the situation of a planar identify the counterion concentration with the monomer surface to which is end-grafted a concentration d-2 of concentration, c. With the use of the step function ideally flexible polymers, each containing N monomeric concentration profile, this immediately yields units. We assume that a fixed fraction f of the monomers possesses an ionic group, e.g., sulfonate or carboxylate. L f'12Nu (11.1) (To be explicit, we suppose that the charges fixed to the The neutralization length 5 is then given by the Debye chain are negative.) This charge fNper chain is neutralized screening length associated with the counterion concenby an equal number of counterions as well as a possible tration, fc. This results in concentration cs of monovalent salt molecules, e.g., NaCl. For simplicity, we only consider the electrostatic inter5 = d(a/4?rZf'/2)'/2 (11.2) actions. This is in contrast to the previous studiesFlswhich Consequently, the condition L 1 5 applies unless N/W4 I also include excluded-volume repulsions. Since our point d(47rZ~)'/~. The result (11.1) is valid except for weakly of view is to elucidate the role of the Coulombic forces, we charged polymers and low grafting densities. Note that do not wish to encumber the results with extraneous the chains are highly stretched for finite charging and, parameters. Furthermore, since the organic backbones furthermore, the brush thickness is independent of the of the polyelectrolytes are typically insoluble in water, it grafting density. would seem more appropriate to use a negative excludedIn the opposite limit, L
